Lecture P9: Trees. Overview. Searching a Database. Searching a Database

Transcription

1 Overview Lecture P9: Trees Culmination of the programming portion of this class. Solve a database searching problem. Trees Versatile and useful data structure. A naturally recursive data structure. Application of stacks and queues. 1/27/00 Copyright 2000, Kevin Wayne P9.1 1/27/00 Copyright 2000, Kevin Wayne P9.2 Searching a Database Searching a Database Database entries. Names and social security numbers. Desired operations. Insert student. Delete student. Search for name given ID number. Goal. All operations fast, even for huge databases. SS# Last Name e Alam Baer Bagyenda Balestri Benjamin Berube Other applications. Online phone book looks up names and telephone numbers. Spell checker looks up words in dictionary. Internet domain server looks up IP addresses. Compiler looks up variable names to find type and memory address. Data structure that supports these operations is called a SYMBOL TABLE. search key 1/27/00 Copyright 2000, Kevin Wayne P9.3 1/27/00 Copyright 2000, Kevin Wayne P9.4

2 Representing the Database Entries Define Item.h file to encapsulate generic database entry. Don t want to use internals of item type when we write database code. want our insert and search code to work for any item type ideally Item would be an ADT Item.c Key is field in search. ITEM.h typedef int Key; typedef struct { Key ID; char name[30]; Item; Item NULLitem = {-1, ""; int eq(key, Key); int less(key, Key); Key key(item); void print(item); #include ITEM.h int eq(key A, Key B) { return A == B; int less(key A, Key B) { return A < B; Key key(item A) { return A.ID; void print(item A) { printf( %9d %20s, A.ID, A.name); Symbol Table ADT Define ST.h file to specify database operations. Make it a true ADT. ST.h (Sedgewick 12.1) Item STsearch(Key); void STinsert(Item); void STprint(void); int STcount(void); void STdelete(Item); 1/27/00 Copyright 2000, Kevin Wayne P9.5 1/27/00 Copyright 2000, Kevin Wayne P9.6 Sorted Array Representation of Database Maintain array of Items. Store in sorted order. Use BINARY SEARCH to find database Item with designated Key. Sorted Array Representation of Database Maintain array of Items. Store in sorted order. Use BINARY SEARCH to find database Item with designated Key. STarray.c (Sedgewick 12.6) Array of database Items. Key k not found. Divide and conquer. #define MAXSIZE Item st[maxsize]; Item search(int l, int r, Key k) { int m = (l+r)/2; if (l > r) return NULLitem; if eq(k, key(st[m])) Key k found. return st[m]; if less(k, key(st[m])) return search(l, m-1, k); return search(m+1, r, k); Wrapper for search function. STarray.c (Sedgewick 12.6) Item STsearch(Key k) { int N = Stcount(); return search(0, N-1, k); 1/27/00 Copyright 2000, Kevin Wayne P9.7 1/27/00 Copyright 2000, Kevin Wayne P9.8

3 Cost of Binary Search Insert Using Sorted Array Representation How many comparisons to find a name in database of size N? log 2 N = number of digits in binary representation of N. Divide list in half each time Problem 1: insertion is slow. Want to keep entries in sorted order. Have to move larger keys over one position to right. The log functions grows very slowly. log 2 (thousand) 10 log 2 (million) 20 log 2 (billion) 30 2 N = x x = log 2 N Demo: inserting 25 into a sorted array. 82 Without binary search (or if unsorted), may need to look at EVERY Item. Savings is enormous for large files. 1/27/00 Copyright 2000, Kevin Wayne P9.9 1/27/00 Copyright 2000, Kevin Wayne P9.10 Insert Using Sorted Array Representation Problem 1: insertion is slow. Want to keep entries in sorted order. Have to move larger keys over one position to right. Exercise: write code for insertion Linked List Representation of Database Keep items in a linked list. Store in sorted order. Insert. Only need to change links. No need to move large amounts of data. STlist.c typedef struct node* link; struct node { Item item; link next; Demo: inserting 25 into a sorted array NULL Problem 2: need to fix maximum database size ahead of time. 25 1/27/00 Copyright 2000, Kevin Wayne P9.11 1/27/00 Copyright 2000, Kevin Wayne P9.12

4 Linked List Representation of Database Keep items in a linked list. Store in sorted order. Insert. Only need to change links. No need to move large amounts of data. STlist.c typedef struct node* link; struct node { Item item; link next; Linked List Representation of Database Search. Can t use binary search since no DIRECT access to middle element. Use sequential search. may need to search entire linked list to find desired Key much slower than binary search NULL 25 STlist.c Item STsearch(Key k) { link x; for (x = head; x!= NULL; x = x->next) if (eq(k, key(x)) return x->item; return NULLitem; 1/27/00 Copyright 2000, Kevin Wayne P9.13 1/27/00 Copyright 2000, Kevin Wayne P9.14 Summary Binary Tree Database entries. Names and social security numbers. Desired operations. Insert, delete, search. Is there any way to have fast insert AND search? Yes. Use TWO links per node. root 14 parent Goal. Make all of these operations FAST even for huge databases. Tradeoff. ARRAY: fast search, slow insert/delete. LINKED LIST: fast insert/delete, slow search. left child right child leaf /27/00 Copyright 2000, Kevin Wayne P9.15 1/27/00 Copyright 2000, Kevin Wayne P9.16

5 Binary Tree in C STbst.h typedef struct STnode* link; struct STnode { Item item; link left; item link right; ; link head; left right Binary Search Tree Binary tree in sorted order. Maintain ordering property for ALL subtrees. root (middle value) Represent in C with two links per node. 66 NULL Leftmost arrow corresponds to left link Rightmost to right link. NULL NULL NULL NULL NULL NULL left subtree (smaller values) right subtree (smaller values) 1/27/00 Copyright 2000, Kevin Wayne P9.17 1/27/00 Copyright 2000, Kevin Wayne P9.18 Binary Search Tree Binary Search Tree Binary tree in sorted order. Maintain ordering property for ALL subtrees. 51 Binary tree in sorted order. Many BST s for the same input data. Have different tree shapes /27/00 Copyright 2000, Kevin Wayne P9.19 1/27/00 Copyright 2000, Kevin Wayne P9.20

6 Search in Binary Search Tree Search in BST s Search for Key k in binary search tree. Analogous to binary search in sorted array. Search for Key k. Search algorithm: Start at head node. If Key of current node is k, return node. Go LEFT if current node has Key < k. Go RIGHT if current node has Key > k. Found Key k. Look for Key k in right subtree. STbst.c (Sedgewick 12.7) Item search (link h, Key k) { if (h == NULL) return NULLitem; Key k not in tree. if (eq(k, key(h->item)) return h->item; if (less(k, key(h->item)) return search(h->left, k); Look for Key k return search(h->right, k); in left subtree. Item STsearch(Key k) { return search(head, k); Search for Key k in BST tree rooted at head. 1/27/00 Copyright 2000, Kevin Wayne P9.21 1/27/00 Copyright 2000, Kevin Wayne P9.22 Cost of BST Search Depends on tree shape. Proportional to length of path from root to Key. Balanced 2 log 2 N comparisons proportional to binary search cost Insert Using BST s How to insert new database Item. Search for key of database Item. Search ends at NULL pointer. New Item belongs here. Allocate memory for new Item, and link it to tree. Unbalanced takes N comparisons for degenerate tree shapes can be as slow as sequential search Algorithm works for any tree shape. With cleverness (see COS 226), can assure tree is always balanced. 1/27/00 Copyright 2000, Kevin Wayne P9.23 1/27/00 Copyright 2000, Kevin Wayne P9.24

7 Insert Using BST s Insertion Cost in BST Allocate memory and insert here. Divide-andconquer. BST.c (Sedgewick 12.7) link insert(link h, Item item) { Key k = key(item); Key k2 = key(h->item); link x; if (h == NULL) { link x = malloc(sizeof *x); x->item = item; x->left = x->right = NULL; return x; if (less(k, k2)) h->left = insert(h->left, item); else h->right = insert(h->right, item); return h; void STinsert(Item item) { head = insert(head, item); Wrapper function. Depends on tree shape. Cost is proportional to length of path from root to node. Tree shape depends on order keys are inserted. Insert in random order. Leads to well-balanced tree average length of path from root to node is 1.44 log 2 N Insert in sorted or reverse-sorted order. degenerates into linked list takes N -1 comparisons With cleverness, can ensure tree is always balanced. see COS 226 1/27/00 Copyright 2000, Kevin Wayne P9.25 1/27/00 Copyright 2000, Kevin Wayne P9.26 Question Other Types of Trees Current code searches for a name given an ID number. What if we want to search for an ID number given a name?! Easy, redefine Item type. Trees. Nodes need not have exactly two children. Order of children may not be important. me Item.h typedef char Key[30]; typedef struct { int ID; Key name; Item; Item NULLitem = {-1, ""; Item.c #include <string.h> int eq(key A, Key B) { return strcmp(a, B) == 0; int less(key A, Key B) { return strcmp(a, B) < 0; Examples. Family tree. mom s mom mom mom s dad dad s mom dad dad s dad int eq(key, Key); int less(key, Key); Key key(item); Key key(item A) { return A.name; 1/27/00 Copyright 2000, Kevin Wayne P9.27 1/27/00 Copyright 2000, Kevin Wayne P9.28

8 Other Types of Trees Trees. Nodes need not have exactly two children. Order of children may not be important. - Examples. Family tree. Parse tree. * + ( a * ( b + c ) ) - ( d + e ) a + d e Trees. Other Types of Trees Nodes need not have exactly two children. Order of children may not be important. Examples. Family tree. Parse tree. Unix file hierarchy. not binary aaclarke files bin lib etc u / cs126 grades zrnye submit mandel stock tsp b c POINT.h point.c tsp13509.txt 1/27/00 Copyright 2000, Kevin Wayne P9.29 1/27/00 Copyright 2000, Kevin Wayne P9.30 Traversing Binary Trees Goal: visit (process) each node in tree in some order. Tree traversal. wrapper function STbst.c void STprint(void) { traverse(head); Traversing Binary Trees Goal: visit (process) each node in tree in some order. Tree traversal. Goal realized no matter what order nodes are visited. inorder: visit between recursive calls traverse left subtree traverse right subtree inorder traverseinorder(link h) { if (h == NULL) return; traverse(h->left); print(h->item); process node h traverse(h->right); inorder traverseinorder(link h) { if (h == NULL) return; traverse(h->left); print(h->item); traverse(h->right); 1/27/00 Copyright 2000, Kevin Wayne P9.31 1/27/00 Copyright 2000, Kevin Wayne P9.32

9 Traversing Binary Trees Traversing Binary Trees Goal: visit (process) each node in tree in some order. Tree traversal. Goal realized no matter what order nodes are visited. inorder: visit between recursive calls preorder: visit before recursive calls Goal: visit (process) each node in tree in some order. Tree traversal. Goal realized no matter what order nodes are visited. inorder: visit between recursive calls preorder: visit before recursive calls postorder: visit after recursive calls preorder traversepreorder(link h) { if (h == NULL) return; print(h->item); traverse(h->left); traverse(h->right); postorder traversepostorder(link h) { if (h == NULL) return; traverse(h->left); traverse(h->right); print(h->item); 1/27/00 Copyright 2000, Kevin Wayne P9.33 1/27/00 Copyright 2000, Kevin Wayne P9.34 Traversing Binary Trees Preorder Traversal With Explicit Stack Goal: visit (process) each node in tree in some order. Tree traversal. Goal realized no matter what order nodes are visited. inorder: visit between recursive calls preorder: visit before recursive calls postorder: visit after recursive calls! Important note: inorder traversal of BST gives free sort! Visit the top node on the stack. Push its children onto stack. Push right node before left, so that left node is visited first. preorder traversal with stack traverse(link h) { STACKpush(h); while (!STACKempty()) { h = STACKpop(); print(h->item); if (h->right!= NULL) STACKpush(h->right); if (h->left!= NULL) STACKpush(h->left); Works for general trees. Generalizes to DEPTH- FIRST-SEARCH in graphs. 1/27/00 Copyright 2000, Kevin Wayne P9.35 1/27/00 Copyright 2000, Kevin Wayne P9.36

10 Level Traversal With Queue Q. What happens if we replace stack with QUEUE? Level order traversal. Visit nodes in order from distance to root. Works for general trees. Generalizes to BREADTH- FIRST-SEARCH in graphs. level traversal with queue traverse(link h) { QUEUEput(h); while (!QUEUEempty()) { h = QUEUEget(); print(h->item); if (h->left!= NULL) QUEUEput(h->left); if(h->right!= NULL) QUEUEput(h->right); Summary How to insert and search a database using: Arrays. Linked lists. Binary search trees. Performance characteristics using different data structures. The meaning of different traversal orders and how the code for them works. 1/27/00 Copyright 2000, Kevin Wayne P9.37 1/27/00 Copyright 2000, Kevin Wayne P9.38